In this paper, we give a new definition for the space of non-holomorphic Jacobi Maas forms (denoted by Jk,mnh) of weight k∈ℤ and index m∈ℕ as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\) . We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in Jk,mnh. We construct new examples of cuspidal Jacobi Maas forms Ff of weight k∈2ℤ and index 1 from weight k−1/2 Maas forms f with respect to Γ0(4) and show that the map f↦Ff is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of Jk,mnh can be “essentially” obtained from scalar or vector valued half integer weight Maas forms.